Editing Rank (linear algebra) (section)

===Proof using row reduction===
The fact that the column and row ranks of any matrix are equal forms is fundamental in linear algebra. Many proofs have been given. One of the most elementary ones has been sketched in {{slink||Rank from row echelon forms}}. Here is a variant of this proof:

It is straightforward to show that neither the row rank nor the column rank are changed by an [[elementary row operation]]. As [[Gaussian elimination]] proceeds by elementary row operations, the [[reduced row echelon form]] of a matrix has the same row rank and the same column rank as the original matrix. Further elementary column operations allow putting the matrix in the form of an [[identity matrix]] possibly bordered by rows and columns of zeros. Again, this changes neither the row rank nor the column rank. It is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries.

We present two other proofs of this result. The first uses only basic properties of [[linear combination]]s of vectors, and is valid over any [[field (mathematics)|field]]. The proof is based upon Wardlaw (2005).<ref name="wardlaw">
{{Citation| last=Wardlaw| first=William P.| title=Row Rank Equals Column Rank| year=2005| journal=[[Mathematics Magazine]]| volume=78| issue=4| pages=316–318| doi=10.1080/0025570X.2005.11953349| s2cid=218542661}}</ref> The second uses [[orthogonality]] and is valid for matrices over the [[real numbers]]; it is based upon Mackiw (1995).<ref name="mackiw" /> Both proofs can be found in the book by Banerjee and Roy (2014).<ref name="banerjee-roy">{{Citation | last1 = Banerjee | first1 = Sudipto | last2 = Roy | first2 = Anindya | date = 2014 | title = Linear Algebra and Matrix Analysis for Statistics | series = Texts in Statistical Science | publisher = Chapman and Hall/CRC | edition = 1st | isbn = 978-1420095388}}</ref>